Math 283 Calculus 3

Spring 2020

Summary

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Summary

  • The main results of this chapter are all higher-dimensional versions of the
  • Fundamental Theorem of Calculus \[ \int_a^b \frac{df}{dt} \;dt = f(b) - f(a) \]
A diagram of 1D line segment


  • An integral of a “derivative” over a region involves the values of the original function only on the orange boundary of the domain

  • Fundamental Theorem for Line Integrals \[ \int_a^b \nabla f \cdot d\mathbf{\vec{r}} = f(\mathbf{\vec{r}}(b)) - f(\mathbf{\vec{r}}(a)) \]
A diagram of a 2D piecewise closed curve

Summary

  • Green’s theorem \[ \iint\limits_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA = \oint\limits_C \mathbf{\vec{F}} \cdot d\mathbf{\vec{r}} \]
A diagram of a piecewise smooth closed curve

  • Stokes' theorem \[ \iint\limits_S (\nabla \times \mathbf{\vec{F}} )\cdot d\mathbf{\vec{S}} = \oint\limits_C \mathbf{\vec{F}} \cdot d\mathbf{\vec{r}} \]
A diagram of a piecewise smooth closed curve

  • Divergence theorem \[ \iiint\limits_E (\nabla \cdot \mathbf{\vec{F}} ) dV = \iint\limits_S \mathbf{\vec{F}} \cdot d\mathbf{\vec{S}} \]
A diagram of a piecewise smooth closed curve

Gauss's Law

vector field flux on sphere

Courtesy of J. Stewart, Calculus: Early transcendentals, 2nd edition

  • Relationship between electric field and electric charges \[ \iint\limits_S \mathbf{\vec{D}}\cdot \mathbf{\hat{N}}dS = \iiint\limits_E \sigma\;dV \]
  • \( \mathbf{\vec{D}} \) electric field density
    \( \sigma \) electric charge density
  • electric density flux around S = total electric charge over E

  • Divergence Theorem \[ \iint\limits_S \mathbf{\vec{D}}\cdot \mathbf{\hat{N}}dS = \iiint\limits_E (\nabla \cdot \mathbf{\vec{D}})\;dV \]
  • Equate \[ \iiint\limits_E (\nabla \cdot \mathbf{\vec{D}})\;dV =\iiint\limits_E \sigma\;dV \]

  • \[ \Leftrightarrow \iiint\limits_E (\sigma- \nabla \cdot \mathbf{\vec{D}})\;dV=0 \]
  • must be true when

    \[ \nabla \cdot \mathbf{\vec{D}} =\sigma \] divergence of electric field density = electric charge density

Gauss's Magnetism Law

  • The magnetic flux of the magnetic field \( \mathbf{\vec{B}} \) out of a closed surface is always zero \[ \iint\limits_S \mathbf{\vec{B}}\cdot d\mathbf{\vec{S}} =0 \]
Circular cylinder
  • Divergence Theorem \[ \iint\limits_S \mathbf{\vec{B}}\cdot d\mathbf{\vec{S}} = \iiint\limits_E (\nabla \cdot \mathbf{\vec{B}})\;dV \]
  • then \[ \iiint\limits_E (\nabla \cdot \mathbf{\vec{B}})\;dV=0 \]
  • must be true when

    \[ \nabla \cdot \mathbf{\vec{B}} =0 \] divergence of magnetic field is zero

Faraday's Law

  • “The line integral of electromotor force around a closed curve equals the negative time rate of change of magnetic flux through a control surface spanning the curve”
  • \[ \oint\limits_C \mathbf{\vec{E}}\cdot d\mathbf{\vec{r}}=-\frac{\partial}{\partial t}(magnetic flux) \]
  • \( \mathbf{\vec{E}} \) electromotor force field
  • \( \mathbf{\vec{B}} \) magnetic field
  • Magnetic flux \[ \hphantom{\oint\limits_C \mathbf{\vec{E}}\cdot d\mathbf{\vec{r}}}=-\frac{\partial}{\partial t}\left(\iint\limits_S \mathbf{\vec{B}}\cdot \mathbf{\hat{N}}dS \right) \]
  • \[ \hphantom{\oint\limits_C \mathbf{\vec{E}} }= -\iint\limits_S \frac{\partial}{\partial t} \mathbf{\vec{B}} \cdot \mathbf{\hat{N}}dS \]
  • Stokes' theorem \[ \oint\limits_C \mathbf{\vec{E}}\cdot d\mathbf{\vec{r}}= \iint\limits_S (\nabla \times \mathbf{\vec{E}})\cdot\:\mathbf{\hat{N}}dS \]

Faraday's Law cont'd

  • Equating \[ \iint\limits_S (\nabla \times \mathbf{\vec{E}})\cdot\:\mathbf{\hat{N}}dS =-\iint\limits_S \frac{\partial}{\partial t} \mathbf{\vec{B}} \cdot \mathbf{\hat{N}}dS \]
  • \[ \hspace{-5cm}\Leftrightarrow \iint\limits_S \left(\nabla \times \mathbf{\vec{E}}+\frac{\partial \mathbf{\vec{B}}}{\partial t} \right) \cdot \mathbf{\hat{N}}dS =0 \]
  • \[ \Leftrightarrow \nabla \times \mathbf{\vec{E}}+\frac{\partial \mathbf{\vec{B}} }{\partial t} = 0 \]

  • \[ \nabla \times \mathbf{\vec{E}}=-\frac{\partial \mathbf{\vec{B}} }{\partial t} \] curl of electric field = - rate of change of magnetic field

Ampere's Law

  • “The current \( I \) passing through a control surface \( S \) equal the line integral of the magnetic field intensity \( \mathbf{\vec{H}} \) around the boundary of \( S \)”
  • \[ current=I=\oint\limits_C \mathbf{\vec{H}}\cdot d\mathbf{\vec{r}} \]
Control surface around a cylindrical cable

Courtesy of J. Stewart, Calculus: Early transcendentals, 2nd edition


  • If our current field is \[ \mathbf{\vec{J}} \]

  • current as the flux of current field through \( S \) \[ I=\iint\limits_S \mathbf{\vec{J}} \cdot \mathbf{\hat{N}}dS \]

Ampere's Law cont'd

Control surface around a cylindrical cable

Courtesy of J. Stewart, Calculus: Early transcendentals, 2nd edition

  • Stokes' theorem:
    \[ current= \oint\limits_C \mathbf{\vec{H}}\cdot d\mathbf{\vec{r}} = \iint\limits_S (\nabla \times \mathbf{\vec{H}})\cdot d\mathbf{\vec{S}} \]
  • But also \[ current=\iint\limits_S \mathbf{\vec{J}} \cdot d\mathbf{\vec{S}} \]
  • Equate \[ \iint\limits_S (\nabla \times \mathbf{\vec{H}})\cdot d\mathbf{\vec{S}} =\iint\limits_S \mathbf{\vec{J}} \cdot d\mathbf{\vec{S}} \]
  • \[ \iint\limits_S \left( \nabla \times \mathbf{\vec{H}} - \mathbf{\vec{J}}\right) \cdot d\mathbf{\vec{S}} = 0 \]

  • \[ \nabla \times \mathbf{\vec{H}} = \mathbf{\vec{J}} \] curl of magnetic field intensity = current field